Entropic value at risk – Wikipedia

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid,^[1][2] which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called “entropic value at risk”. The EVaR was developed to tackle some computational inefficiencies^{[clarification needed]} of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid^[1][2] developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.

Definition[edit]

Let

be a probability space with

a set of all simple events,

a

-algebra of subsets of

and

a probability measure on

. Let

be a random variable and

be the set of all Borel measurable functions

whose moment-generating function

exists for all

. The entropic value at risk (EVaR) of

with confidence level

is defined as follows:

${displaystyle {text{EVaR}}_{1-alpha }(X):=inf _{z>0}left{z^{-1}ln left({frac {M_{X}(z)}{alpha }}right)right}.}$

(1)

In finance, the random variable

in the above equation, is used to model the losses of a portfolio.

Consider the Chernoff inequality

${displaystyle Pr(Xgeq a)leq e^{-za}M_{X}(z),quad forall z>0.}$

(2)

Solving the equation

for

results in

${displaystyle a_{X}(alpha ,z):=z^{-1}ln left({frac {M_{X}(z)}{alpha }}right).}$

By considering the equation (1), we see that

${displaystyle {text{EVaR}}_{1-alpha }(X):=inf _{z>0}{a_{X}(alpha ,z)},}$

${displaystyle a_{X}(1,z)}$

is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively.

Let

${displaystyle mathbf {L} _{M}}$

be the set of all Borel measurable functions

${displaystyle X:Omega to mathbb {R} }$

whose moment-generating function

${displaystyle M_{X}(z)}$

exists for all

${displaystyle z}$

. The dual representation (or robust representation) of the EVaR is as follows:

${displaystyle {text{EVaR}}_{1-alpha }(X)=sup _{Qin Im }(E_{Q}(X)),}$

(3)

where

${displaystyle Xin mathbf {L} _{M},}$

and

${displaystyle Im }$

is a set of probability measures on

${displaystyle (Omega ,{mathcal {F}})}$

with

${displaystyle Im ={Qll P:D_{KL}(Q||P)leq -ln alpha }}$

. Note that

${displaystyle D_{KL}(Q||P):=int {frac {dQ}{dP}}left(ln {frac {dQ}{dP}}right)dP}$

is the relative entropy of

${displaystyle Q}$

with respect to

${displaystyle P,}$

also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.

Properties[edit]

• The EVaR is a coherent risk measure.

• The moment-generating function
  ${displaystyle M_{X}(z)}$

  can be represented by the EVaR: for all

  ${displaystyle Xin mathbf {L} _{M^{+}}}$

  and

  ${displaystyle z>0}$

  

  ${displaystyle M_{X}(z)=sup _{0$

  

  (4)

• The entropic risk measure with parameter
  ${displaystyle theta ,}$

  can be represented by means of the EVaR: for all

  ${displaystyle Xin mathbf {L} _{M^{+}}}$

  and

  ${displaystyle theta >0}$

  

  ${displaystyle theta ^{-1}ln M_{X}(theta )=a_{X}(1,theta )=sup _{0$

  

  (5)

  • The EVaR with confidence level
    ${displaystyle 1-alpha }$

    is the tightest possible upper bound that can be obtained from the Chernoff inequality for the VaR and the CVaR with confidence level

    ${displaystyle 1-alpha }$

    ;

  ${displaystyle {text{VaR}}(X)leq {text{CVaR}}(X)leq {text{EVaR}}(X).}$

  (6)

  • The following inequality holds for the EVaR:

  ${displaystyle {text{E}}(X)leq {text{EVaR}}_{1-alpha }(X)leq {text{esssup}}(X)}$

  (7)

  where

  ${displaystyle {text{E}}(X)}$

  is the expected value of

  ${displaystyle X}$

  and

  ${displaystyle {text{esssup}}(X)}$

  is the essential supremum of

  ${displaystyle X}$

  , i.e.,

  ${displaystyle inf _{tin mathbb {R} }{t:Pr(Xleq t)=1}}$

  . So do hold

  ${displaystyle {text{EVaR}}_{0}(X)={text{E}}(X)}$

  and

  ${displaystyle lim _{alpha to 0}{text{EVaR}}_{1-alpha }(X)={text{esssup}}(X)}$

  .

  Examples[edit]

  

  Comparing the VaR, CVaR and EVaR for the standard normal distribution
  

  Comparing the VaR, CVaR and EVaR for the uniform distribution over the interval (0,1)

  For

  ${displaystyle Xsim N(mu ,sigma ^{2}),}$

  

  ${displaystyle {text{EVaR}}_{1-alpha }(X)=mu +sigma {sqrt {-2ln alpha }}.}$

  (8)

  For

  ${displaystyle Xsim U(a,b),}$

  

  ${displaystyle {text{EVaR}}_{1-alpha }(X)=inf _{t>0}leftlbrace tln left(t{frac {e^{t^{-1}b}-e^{t^{-1}a}}{b-a}}right)-tln alpha rightrbrace .}$

  (9)

  Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for

  ${displaystyle N(0,1)}$

  and

  ${displaystyle U(0,1)}$

  .

  Optimization[edit]

  Let

  ${displaystyle rho }$

  be a risk measure. Consider the optimization problem

  ${displaystyle min _{{boldsymbol {w}}in {boldsymbol {W}}}rho (G({boldsymbol {w}},{boldsymbol {psi }})),}$

  (10)

  where

  ${displaystyle {boldsymbol {w}}in {boldsymbol {W}}subseteq mathbb {R} ^{n}}$

  is an

  ${displaystyle n}$

  -dimensional real decision vector,

  ${displaystyle {boldsymbol {psi }}}$

  is an

  ${displaystyle m}$

  -dimensional real random vector with a known probability distribution and the function

  ${displaystyle G({boldsymbol {w}},.):mathbb {R} ^{m}to mathbb {R} }$

  is a Borel measurable function for all values

  ${displaystyle {boldsymbol {w}}in {boldsymbol {W}}.}$

  If

  ${displaystyle rho ={text{EVaR}},}$

  then the optimization problem (10) turns into:

  ${displaystyle min _{{boldsymbol {w}}in {boldsymbol {W}},t>0}left{tln M_{G({boldsymbol {w}},{boldsymbol {psi }})}(t^{-1})-tln alpha right}.}$

  (11)

  Let

  ${displaystyle {boldsymbol {S}}_{boldsymbol {psi }}}$

  be the support of the random vector

  ${displaystyle {boldsymbol {psi }}.}$

  If

  ${displaystyle G(.,{boldsymbol {s}})}$

  is convex for all

  ${displaystyle {boldsymbol {s}}in {boldsymbol {S}}_{boldsymbol {psi }}}$

  , then the objective function of the problem (11) is also convex. If

  ${displaystyle G({boldsymbol {w}},{boldsymbol {psi }})}$

  has the form

  ${displaystyle G({boldsymbol {w}},{boldsymbol {psi }})=g_{0}({boldsymbol {w}})+sum _{i=1}^{m}g_{i}({boldsymbol {w}})psi _{i},qquad g_{i}:mathbb {R} ^{n}to mathbb {R} ,i=0,1,ldots ,m,}$

  (12)

  and

  ${displaystyle psi _{1},ldots ,psi _{m}}$

  are independent random variables in

  ${displaystyle mathbf {L} _{M}}$

  , then (11) becomes

  ${displaystyle min _{{boldsymbol {w}}in {boldsymbol {W}},t>0}leftlbrace g_{0}({boldsymbol {w}})+tsum _{i=1}^{m}ln M_{g_{i}({boldsymbol {w}})psi _{i}}(t^{-1})-tln alpha rightrbrace .}$

  (13)

  which is computationally tractable. But for this case, if one uses the CVaR in problem (10), then the resulting problem becomes as follows:

  ${displaystyle min _{{boldsymbol {w}}in {boldsymbol {W}},tin mathbb {R} }leftlbrace t+{frac {1}{alpha }}{text{E}}left[g_{0}({boldsymbol {w}})+sum _{i=1}^{m}g_{i}({boldsymbol {w}})psi _{i}-tright]_{+}rightrbrace .}$

  (14)

  It can be shown that by increasing the dimension of

  ${displaystyle psi }$

  , problem (14) is computationally intractable even for simple cases. For example, assume that

  ${displaystyle psi _{1},ldots ,psi _{m}}$

  are independent discrete random variables that take

  ${displaystyle k}$

  distinct values. For fixed values of

  ${displaystyle {boldsymbol {w}}}$

  and

  ${displaystyle t,}$

  the complexity of computing the objective function given in problem (13) is of order

  ${displaystyle mk}$

  while the computing time for the objective function of problem (14) is of order

  ${displaystyle k^{m}}$

  . For illustration, assume that

  ${displaystyle k=2,m=100}$

  and the summation of two numbers takes

  ${displaystyle 10^{-12}}$

  seconds. For computing the objective function of problem (14) one needs about

  ${displaystyle 4times 10^{10}}$

  years, whereas the evaluation of objective function of problem (13) takes about

  ${displaystyle 10^{-10}}$

  seconds. This shows that formulation with the EVaR outperforms the formulation with the CVaR (see ^[2] for more details).

  Generalization (g-entropic risk measures)[edit]

  Drawing inspiration from the dual representation of the EVaR given in (3), one can define a wide class of information-theoretic coherent risk measures, which are introduced in.^[1][2] Let

  ${displaystyle g}$

  be a convex proper function with

  ${displaystyle g(1)=0}$

  and

  ${displaystyle beta }$

  be a non-negative number. The

  ${displaystyle g}$

  -entropic risk measure with divergence level

  ${displaystyle beta }$

  is defined as

  ${displaystyle {text{ER}}_{g,beta }(X):=sup _{Qin Im }{text{E}}_{Q}(X)}$

  (15)

  where

  ${displaystyle Im ={Qll P:H_{g}(P,Q)leq beta }}$

  in which

  ${displaystyle H_{g}(P,Q)}$

  is the generalized relative entropy of

  ${displaystyle Q}$

  with respect to

  ${displaystyle P}$

  . A primal representation of the class of

  ${displaystyle g}$

  -entropic risk measures can be obtained as follows:

  ${displaystyle {text{ER}}_{g,beta }(X)=inf _{t>0,mu in mathbb {R} }leftlbrace tleft[mu +{text{E}}_{P}left(g^{*}left({frac {X}{t}}-mu +beta right)right)right]rightrbrace }$

  (16)

  where

  ${displaystyle g^{*}}$

  is the conjugate of

  ${displaystyle g}$

  . By considering

  ${displaystyle g(x)={begin{cases}xln x&x>0\0&x=0\+infty &x<0end{cases}}}$

  (17)

  with

  ${displaystyle g^{*}(x)=e^{x-1}}$

  and

  ${displaystyle beta =-ln alpha }$

  , the EVaR formula can be deduced. The CVaR is also a

  ${displaystyle g}$

  -entropic risk measure, which can be obtained from (16) by setting

  ${displaystyle g(x)={begin{cases}0&0leq xleq {frac {1}{alpha }}\+infty &{text{otherwise}}end{cases}}}$

  (18)

  with

  ${displaystyle g^{*}(x)={tfrac {1}{alpha }}max{0,x}}$

  and

  ${displaystyle beta =0}$

  (see ^[1][3] for more details).

  For more results on

  ${displaystyle g}$

  -entropic risk measures see.^[4]

  Disciplined Convex Programming Framework[edit]

  The disciplined convex programming framework of sample EVaR was proposed by Cajas^[5] and has the following form:

  ${displaystyle {begin{aligned}{text{EVaR}}_{alpha }(X)&=left{{begin{array}{ll}{underset {z,,t,,u}{text{min}}}&t+zln left({frac {1}{alpha T}}right)\{text{s.t.}}&zgeq sum _{j=1}^{T}u_{j}\&(X_{j}-t,z,u_{j})in K_{text{exp}};forall ;j=1,ldots ,T\end{array}}right.end{aligned}}}$

  (19)

  where

  ${displaystyle z}$

  ,

  ${displaystyle t}$

  and

  ${displaystyle u}$

  are variables;

  ${displaystyle K_{text{exp}}}$

  is an exponential cone;^[6] and

  ${displaystyle T}$

  is the number of observations. If we define

  ${displaystyle w}$

  as the vector of weights for

  ${displaystyle N}$

  assets,

  ${displaystyle r}$

  the matrix of returns and

  ${displaystyle mu }$

  the mean vector of assets, we can posed the minimization of the expected EVaR given a level of expected portfolio return

  ${displaystyle {bar {mu }}}$

  as follows.

  ${displaystyle {begin{aligned}&{underset {w,,z,,t,,u}{text{min}}}&&t+zln left({frac {1}{alpha T}}right)\&{text{s.t.}}&&mu w^{tau }geq {bar {mu }}\&&&sum _{i=1}^{N}w_{i}=1\&&&zgeq sum _{j=1}^{T}u_{j}\&&&(-r_{j}w^{tau }-t,z,u_{j})in K_{text{exp}};forall ;j=1,ldots ,T\&&&w_{i}geq 0;;;forall ;i=1,ldots ,N\end{aligned}}}$

  (20)

  Applying the disciplined convex programming framework of EVaR to uncompounded cumulative returns distribution, Cajas^[5] proposed the entropic drawdown at risk(EDaR) optimization problem. We can posed the minimization of the expected EDaR given a level of expected return

  ${displaystyle {bar {mu }}}$

  as follows:

  ${displaystyle {begin{aligned}&{underset {w,,z,,t,,u,,d}{text{min}}}&&t+zln left({frac {1}{alpha T}}right)\&{text{s.t.}}&&mu w^{tau }geq {bar {mu }}\&&&sum _{i=1}^{N}w_{i}=1\&&&zgeq sum _{j=1}^{T}u_{j}\&&&(d_{j}-R_{j}w^{tau }-t,z,u_{j})in K_{text{exp}};forall ;j=1,ldots ,T\&&&d_{j}geq R_{j}w^{tau };forall ;j=1,ldots ,T\&&&d_{j}geq d_{j-1};forall ;j=1,ldots ,T\&&&d_{j}geq 0;forall ;j=1,ldots ,T\&&&d_{0}=0\&&&w_{i}geq 0;;;forall ;i=1,ldots ,N\end{aligned}}}$

  (21)

  where

  ${displaystyle d}$

  is a variable that represent the uncompounded cumulative returns of portfolio and

  ${displaystyle R}$

  is the matrix of uncompounded cumulative returns of assets.

  For other problems like risk parity, maximization of return/risk ratio or constraints on maximum risk levels for EVaR and EDaR, you can see ^[5] for more details.

  The advantage of model EVaR and EDaR using a disciplined convex programming framework, is that we can use softwares like CVXPY ^[7] or MOSEK^[8] to model this portfolio optimization problems. EVaR and EDaR are implemented in the python package Riskfolio-Lib.^[9]

  See also[edit]

  References[edit]